The **main objective** of this question is to find the complement measure for the given statement.

This question uses the concept of **complementary angle** and **complement measure**. Two angles are said to be **complementary** if their **sum** results in **90** **degrees,** and for **complement measure** we have this **formula**:

** 90 – x**

## Expert Answer

We have to find the **complement measure,** which is **mathematically** equal to:

\[90 \space – \space x \]

From the **given statement**, we know that:

\[x \space = \space 5 (90 \space – \space x ) \space – \space 6 \]

We have to **solve** it for $ x $, results in:

\[x \space = \space 450 \space – \space 5 x \space – \space 6 \]

**Subtracting** $ 6 $ from $ 450 $results in:

\[x \space = \space 444 \space – \space 5 x \]

**Adding** $ 5x $ to both sides results in:

\[6x \space = \space 444 \]

**Dividing** by $ 6 $ on both sides results in:

\[x \space = \space 74 \]

Now we know that the **complement measure** is:

\[90 \space – \space x \]

**So**:

\[= \space 90 \space – \space 74 \]

\[= \space 16 ^ {\circ} \].

## Numerical Answer

The **complement measure** for the **given statement** is $ 16 ^ {\circ} $.

## Example

Determine the complement measure so the measure angle becomes 8 less and 10 less than six times of its complement.

We have to find the** complement measure** which is **mathematically** equal to:

\[90 \space – \space x \]

From the **given statement**, we know that:

\[x \space = \space 6 (90 \space – \space x ) \space – \space 8 \]

We have to **solve** it for $ x $, resulting in:

\[x \space = \space 540 \space – \space 6 x \space – \space 8 \]

**Subtracting** $ 8 $ from $ 540 $results in:

\[x \space = \space 532 \space – \space 6 x \]

**Adding** $ 6x $ to both sides results in:

\[7x \space = \space 532 \]

**Dividing** by $ 7 $ on both sides results in:

\[x \space = \space 76 \]

Now we know that the **complement measure** is:

\[90 \space – \space x \]

**So**:

\[= \space 90 \space – \space 76 \]

\[= \space 14 ^ {\circ} \].

**Now**:

We have to find the **complement measure,** which is **mathematically** equal to:

\[90 \space – \space x \]

From the **given statement**, we know that:

\[x \space = \space 6 (90 \space – \space x ) \space – \space 10 \]

We have to solve it for $ x $, **resulting** in:

\[x \space = \space 540 \space – \space 6 x \space – \space 10 \]

**Subtracting** $ 8 $ from $ 540 $results in:

\[x \space = \space 530 \space – \space 6 x \]

**Adding** $ 6x $ to both sides **results** in:

\[7x \space = \space 530 \]

Dividing by $ 7 $ on **both sides** results in:

\[x \space = \space 75.71 \]

Now we know that the **complement measure** is:

\[90 \space – \space x \]

**So**:

\[= \space 90 \space – \space 75.71 \]

\[= \space 14.29 ^ {\circ} \].